[论文解读] Distinguishing quantum states using Clifford orbits

Richard Kueng, Huangjun Zhu+1 more|arXiv (Cornell University)|Sep 27, 2016

Quantum Computing Algorithms and Architecture被引用 13

一句话总结

本文研究了基于 Clifford 群轨道的测量对量子态的可区分性，表明由 Clifford 群导出的 stabilizer 测量在区分纯量子态方面几乎是最佳的。关键贡献在于，基于态的有效秩和相空间局域性，给出了 POVM 范数常数的定量界，且 stabilizer 测量在纯态情况下实现了近乎最优的性能。

ABSTRACT

It is a fundamental property of quantum mechanics that information is lost as a result of performing measurements. Indeed, with every quantum measurement one can associate a number -- its POVM norm constant -- that quantifies how much the distinguishability of quantum states degrades in the worst case as a result of the measurement. This raises the obvious question which measurements preserve the most information in these sense of having the largest norm constant. While a number of near-optimal schemes have been found (e.g. the uniform POVM, or complex projective 4-designs), they all seem to be difficult to implement in practice. Here, we analyze the distinguishability of quantum states under measurements that are orbits of the Clifford group. The Clifford group plays an important role e.g. in quantum error correction, and its elements are considered simple to implement. We find that the POVM norm constants of Clifford orbits depend on the effective rank of the states that should be distinguished, as well as on a quantitative measure of the "degree of localization in phase space" of the vectors in the orbit. The most important Clifford orbit is formed by the set of stabilizer states. Our main result implies that stabilizer measurements are essentially optimal for distinguishing pure quantum states. As an auxiliary result, we use the methods developed here to prove new entropic uncertainty relations for stabilizer measurements. This paper is based on a very recent analysis of the representation theory of tensor powers of the Clifford group.

研究动机与目标

• 确定哪些量子测量在最坏情况下能最大程度地保持量子态的可区分性。
• 分析由 Clifford 群轨道构成的测量的性能，这些测量在量子信息中具有实际可实现性且被广泛使用。
• 利用 POVM 范数常数量化 Clifford 轨道 POVM 在多大程度上保持了量子态的可区分性。
• 建立 stabilizer 测量——Clifford 轨道的特例——在区分纯量子态方面本质上是最优的。
• 利用所提出的框架推导 stabilizer 测量的新熵不确定关系。

提出的方法

• 使用 POVM 范数常数作为衡量标准，量化测量在最坏情况下保持态可区分性的能力。
• 应用 Clifford 群张量幂的表示理论，分析 Clifford 轨道 POVM 的行为。
• 引入矩阵的“有效秩”概念，以改进对可区分性的界，特别是针对纯态。
• 利用 Pinsker 不等式推导量子测量中互信息的确定性关系（熵下界）。
• 通过状态差的迹范数，推导测量输出 ℓ1-范数的界，将其与 POVM 范数常数联系起来。
• 考虑纯态的各向同性系综，并分析制备与测量结果之间的互信息。

实验结果

研究问题

• RQ1 Clifford 轨道测量在多大程度上保持了量子态的可区分性，尤其是在最坏情况下？
• RQ2 量子态的有效秩与 Clifford 轨道 POVM 在区分该态时的性能之间有何关系？
• RQ3 Clifford 轨道中向量的相空间局域性如何影响 POVM 范数常数？
• RQ4 Stabilizer 测量能否被视为区分纯量子态的最优测量，若可以，其条件是什么？
• RQ5 从 Clifford 轨道 POVM 的分析中可以推导出哪些新的熵不确定关系？

主要发现

• Clifford 轨道测量的 POVM 范数常数取决于态的有效秩及其相空间局域性，且 stabilizer 态实现了近乎最优的性能。
• 对于纯态，stabilizer 测量的 POVM 范数常数满足 ‖ℳstab(X)‖₁ ≥ (1/2)‖X‖₁，与均匀 POVM 的最优界一致。
• 该 stabilizer 测量的界通过 X = ρ − (1/d)𝕀 的有效秩 r_eff(X) ≤ 4 min{r, d−r} 推导得出，且在纯态时达到饱和。
• 建立了新的确定性关系：对任意各向同性纯态系综，有 I(X:ℳC,z) ≥ (1/(128 log 2))((d−1)/d)²，该结果对所有 Clifford POVM 均成立。
• 该界同样适用于测量结果的最小熵，即 I(X:ℳC,z) ≥ (1/(128 log 2))((d−1)/d)²，优于先前结果。
• 该分析导出了 stabilizer 测量的新熵不确定关系，其最小熵下界为 (1/(128 log 2))((d−1)/d)²（对纯态）。